How to interpolate PDF(Probability Distribution Functions) from CDF (without root finding method) ?
Please tell the steps to do so.
Thanks.
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1 Answer
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If you have an analytical form of the CDF, you can simply take the first derivative to obtain the PDF (for a continuous distribution). If you have numerical data points representing a CDF, you can construct a numerical approximation to the first derivative by using a finite difference method. If you're going the numerical route, you should use at least a symmetric second-order finite difference, since it's just as easy as the first-order methods.
$$ PDF(x) = \frac{CDF(x+\delta x) - CDF(x-\delta x)}{2 \,\delta x} + O(\delta x^2) $$
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$\begingroup$ In case of Numerical data points ,Suppose I have data set at a & b. So, you mean I need to calculate the differential using FD method at a& b and then interpolate for all the points in between a& b? But to calculate it , we should have close data points near a & b for CDF ? $\endgroup$– quant1Oct 31, 2015 at 14:35
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$\begingroup$ If you have the values of the CDF at $x=a$ and $x=b$, you will be able to most accurately calculate the value of the PDF at $x=(a+b)/2$. If you want the PDF at other locations, you will have to settle for lower accuracy or use other values of the CDF. If I were you, I'd look into how and why finite difference methods work prior to applying them. If you're willing to go out-of-field for the info, every book on numerical fluid dynamics has a very thorough treatment of finite difference methods, and how to construct them for a desired level of accuracy. $\endgroup$ Oct 31, 2015 at 15:15
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$\begingroup$ You are right. I read somewhere that in order to carry out the FD scheme in the most efficient way , the denominator delta(x) should be equal to (12*y *e/y'''')^0.25 .Here y in numerator = CDF() y'''' is the 4th order derivative; e being a suitable presentation of machine precision. Can you please mention how to estimate e ? $\endgroup$– quant1Nov 1, 2015 at 7:46
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$\begingroup$ I haven't seen that formula before, and I suspect that it may have come from a slightly different finite difference method since 4th derivatives are not involved in the error term for this one. That being said, the machine epsilon is a measure of the precision used to store a number using a floating-point representation on a computer. For 64-bit floating-point numbers (default in matlab, octave, ...), you will get ~15-16 significant decimal digits (52 binary digits). Thus, the machine epsilon is $2^{-52}$. The wikipedia page has a more thorough explanation than can fit in a comment. $\endgroup$ Nov 1, 2015 at 19:36
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$\begingroup$ As a practical matter, your data will not have sufficient resolution to use the formula that you listed. You should use the 2 data points centered around the point of interest to compute your approximation of the derivative. Your formula will likely predict a $\delta x < 10^{-15}$. $\endgroup$ Nov 1, 2015 at 19:44